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This paper discusses consensus control for a kind of dynamical agents in network. It is assumed that the agents distributed on a plane and their location coordinates are measured by remote sensor and transmitted to its neighbors. By designing the linear distributed control protocol, it is shown that the group of agents will achieves consensus. The simulations are given to show the effectiveness of our theoretical result.

Distributed coordination of network of dynamic agents has attracted a great attention in recent years. Modeling and exploring these coordinated dynamic agents have become an important issue in physics, biophysics, systems biology, applied mathematics, mechanics, computer science and control theory [

To describe the collective behavior of agents in a large scale network, the agent in the network usually is modeled by a very simple mathematical model, which is an approximation of real objects. Saber and Murray [

In our work a similar problem is studied under the condition that the agents move in a plane. The agents may represent the vehicles or mobile robots spread over a wild area and they communicate by means of some remote sensors with certain error. When the agents are moving in a plane, the collective behavior conditions will depend on the communicated error and the algebraic characterization of the communicated network topology, as well as the dynamical behavior of agents.

This paper is organized as follows. In Section 2, we recall some properties on graph theory and give the problem formulation. In Section 3 the main results of this paper are given and some simulation results are presented in Section 4. Final section is a conclusion.

Consider a network of dynamical agents defined by a graph

Let

where

servation matrix of the agent by some remote sensor.

In what follows we simply assume that

neighbors through the network. The matrix C is assumed to be an orthogonal matrix in the form

The parameter

For the dynamic agent (1) in network we have following assumption.

Assumption 2.1 The dynamics (1) is Lyapunov stable when it disconnected with its neighbors, meaning that the dynamical agent as an autonomous will gradually stop by moving a finite distance for any non-zero initial velocity

The collective behavior of dynamical agents in network can be described by

In this work, we discuss the collective behavior of the dynamical agents under a decentralized control law in the form that

where indexes

We claim that a group of dynamical agents associated with

In our work, let (2) be

where

Remark 1: If we choose

for

Consider a group of dynamical agents in network associated with a graph

Denote

where

Let

where

and L is the aforementioned Laplacian associated with the graph

The collective behavior problem of dynamical agents can be described in

As dynamics (7) is a standard linear time-invariant dynamical system, its trajectory can be described by

The consensus asymptotical stability implies that the matrix

Lemma 3.1 The matrix

and

Proof: It is well known that the graph

Thus,

The following Lemma is key to our work.

Lemma 3.2 If the control gain k in dynamical agent (1) satisfies Assumption 2.1, and

with

where

Proof: Denote the eigenvalues of L by

One can verify the following formulae.

The dynamical behavior of the network (7) is characterized by the eigenvalues of

First we discuss the block with

For

Consider the characteristic polynomial of

where

Construct the Routh array of

with

criterion, for stability it is necessary that

By (14) one has

and

The inequalities (15) can be rewritten as the following form by using the conditions of Lemma 3.2 and the Equations (16)-(17).

We can further show that the second inequality in above implies the first one. Obviously, it is true when

where

Thus, one can consider the following inequalities

The last inequality obviously holds. Therefore, the solution of (18) leads

If

computing process. It shows that

Therefore,

By

where

Let

correspondingly.

As

and

Let

Due to the fact that

Theorem 3.1 Under conditions of Lemma 3.2 the control protocol (4) globally and asymptotically achieves the collective behavior of the dynamic agents.

Proof: As

Therefore,

and it is obvious that

This implies the protocol (5) globally asymptotically achieve aggregation.

Corollary 3.1 If the control gain k satisfies

Under Assumption 2.1 one has

Corollary 3.2 The dynamical agents achieve collective behavior if

We study some examples to show that our results are effective. The network of dynamic agents is described in

We can obtain the Laplacian matrix L of the graph

We consider that the dynamic agent (1) in the network has

When a control protocol (4) is applied into the agents in network, the collective behavior of dynamic agents takes place according to our result.

It is found that when the agents approach to

We discuss the consensus control of dynamical agents in network which associated with a graph

agents are moving in a plane, the aggregation of the dynamical agents are depended on not only the communicated error, but also the algebraic characterization of the communicated network graph and the dynamical properties of agents.

This work was supported by the Natural Science Foundation of Jiangsu Higher Education Institutions of China (Grant no. 13KJB110015).

HongwangYu, (2016) Consensus Control for a Kind of Dynamical Agents in Network. International Journal of Communications, Network and System Sciences,09,29-37. doi: 10.4236/ijcns.2016.91003